Error Analysis of some Finite Element Methods for the Stokes Problem

Abstract

The author proves the optimal convergence for some two-dimensional finite element methods for the Stokes equations. First two families of `Taylor- Hood' type methods: the triangular P\sb 3-P\sb 2 element and the Q\sb k-Q\sb{k-1}, k\ge 2, family of quadrilateral elements are considered. Finally two new low-order methods with piecewise constant approximations for the pressure are introduced and analyzed. Macro-element technique, introduced by the author [ibid. 42, 9-23 (1984; Zbl 0535.76037)] is used in a slightly more practical form for error analysis. \par It is claimed that the results of the paper are trivially also valid when the same finite element spaces are used for equations of (nearly) incompressible elasticity. The author admits that some of the results presented herein have also been obtained by {\it F. Brezzi} and {\it R. S. Falk} [Stability of a higher order Hood-Taylor method, SIAM, J. Numer. Anal. (to appear)].